A Sobolev space theory for the stochastic partial differential equations with space-time non-local operators

We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes ∂tαu=(ϕ(Δ)u+f(u))+∂tβ∑k=1∞∫0tgk(u)dwsk,t>0,x∈Rdas well as the SPDE driven by space-time white noise ∂tαu=ϕ(Δ)u+f(u)+∂tβ-1h(u)W˙,t>0,x∈Rd.Here, α∈ (0 , 1) , β< α+ 1 / 2 , {wtk:k=1,2,…} is a family of independent one-dimensional Wiener processes and W˙ is a space-time white noise defined on [0 , ∞) × R^d. The time non-local operator ∂tα denotes the Caputo fractional derivative of order α, the function ϕ is a Bernstein function, and the spatial non-local operator ϕ(Δ) is the integro-differential operator whose symbol is - ϕ(| ξ| ²). In other words, ϕ(Δ) is the infinitesimal generator of the d-dimensional subordinate Brownian motion. We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity results of solutions.